CLASSICAL METHODS IN THE THEORY OF LATTICE PACKINGS
S. S. Ryshkov, E. P. Baranovskii · Russian Mathematical Surveys · 1979
ContentsChapter I. Introduction ??1. The objectives of the article ??2. Positive quadratic forms and point lattices ??3. The arithmetical minimum of positive quadratic forms. Lattice packingsChapter II. The methods of Korkine and Zolotarev ??4. The result of the first memoir of Korkine and Zolotarev and its generalizations; the theorems of Mordell and Rankin ??5. The reduction of Korkine and Zolotarev ??6. Conditions that the coefficients in a Korkine-Zolotarev expansion must satisfy. The densest lattice packings for n = 2, 3, 4 ??7. Extreme forms. The densest lattice packing for n = 5. The investigations of Watson. The results of Stacey ??8. The investigations of Blichfeldt (with refinements announced by Watson and proofs by Vetchinkin)Chapter III. The geometry of the space of coefficients and the Voronoi polyhedron ??9. The cone of positivity and some groups of automorphisms of it ??10. Equidiscriminant surfaces. Three types of planes related to the cone K ??11. Voronoi's polyhedron: definition and elementary properties ??12. The (N ? 1)-dimensional faces of the Voronoi polyhedron and perfect forms ??13. The theorem of Korkine and Zolotarev and the theorem of Voronoi about extreme and perfect forms ??14. An algorithm for constructing the Voronoi polyhedron. The polyhedron ?(n) for n = 2 and?3 ??15. The second and third perfect faces of the Voronoi polyhedron. The polyhedra ?(4) and ?(5) ??16. Results on ?(n) for n ? 6. Applications of the Voronoi polyhedron and other problemsChapter IV. On the Minkowski theory ??17. The Minkowski reduction ??18. Edge formsReferences